View source for Liouville's theorem (differential algebra)

{{short description|Says when antiderivatives of elementary functions can be expressed as elementary functions}}
In [[mathematics]], '''Liouville's theorem''', originally formulated by French mathematician [[Joseph Liouville]] in 1833 to 1841,<ref>{{harvnb|Liouville|1833a}}.</ref><ref>{{harvnb|Liouville|1833b}}.</ref><ref>{{harvnb|Liouville|1833c}}.</ref> places an important restriction on [[antiderivative]]s that can be expressed as [[elementary function]]s.

The antiderivatives of certain [[elementary function]]s cannot themselves be expressed as elementary functions.  These are called [[Nonelementary integral|nonelementary antiderivative]]s. A standard example of such a function is <math>e^{-x^2},</math> whose antiderivative is (with a multiplier of a constant) the [[error function]], familiar from [[statistics]].  Other examples include the functions <math>\frac{\sin (x)}{x}</math> and <math>x^x.</math>

Liouville's theorem states that elementary antiderivatives, if they exist, are in the same [[differential field]] as the function, plus possibly a finite number of applications of the logarithm function.

==Definitions==

For any [[differential field]] <math>F,</math> the '''[[Mathematical constant|{{visible anchor|Constants of a differential field|text=constants}}]]''' of <math>F</math> is the subfield
<math display=block>\operatorname{Con}(F) = \{ f \in F : D f = 0\}.</math>
Given two differential fields <math>F</math> and <math>G,</math> <math>G</math> is called a '''{{visible anchor|logarithmic extension}}''' of <math>F</math> if <math>G</math> is a [[field extension|simple transcendental extension]] of <math>F</math> (that is, <math>G = F(t)</math> for some [[Transcendental element|transcendental]] <math>t</math>) such that
<math display=block>D t = \frac{D s}{s} \quad \text{ for some } s \in F.</math>

This has the form of a [[logarithmic derivative]]. Intuitively, one may think of <math>t</math> as the [[logarithm]] of some element <math>s</math> of <math>F,</math> in which case, this condition is analogous to the ordinary [[chain rule]].  However, <math>F</math> is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to <math>F.</math>  Similarly, an '''{{visible anchor|exponential extension}}''' is a simple transcendental extension that satisfies
<math display=block> \frac{D t}{t} = D s \quad \text{ for some } s \in F.</math>

With the above caveat in mind, this element may be thought of as an exponential of an element <math>s</math> of <math>F.</math>  Finally, <math>G</math> is called an '''{{visible anchor|elementary differential extension}}''' of <math>F</math> if there is a finite chain of [[Subfield (mathematics)|subfields]] from <math>F</math> to <math>G</math> where each [[Field extension|extension]] in the chain is either algebraic, logarithmic, or exponential.

==Basic theorem==

Suppose <math>F</math> and <math>G</math> are differential fields with <math>\operatorname{Con}(F) = \operatorname{Con}(G),</math> and that <math>G</math> is an [[#elementary differential extension|elementary differential extension]] of <math>F.</math> Suppose <math>f \in F</math> and <math>g \in G</math> satisfy <math>D g = f</math> (in words, suppose that <math>G</math> contains an antiderivative of <math>f</math>). 
Then there exist <math>c_1, \ldots, c_n \in \operatorname{Con}(F)</math> and <math>f_1, \ldots, f_n, s \in F</math> such that
<math display=block>f = c_1 \frac{D f_1}{f_1} + \dotsb + c_n \frac{D f_n}{f_n} + D s.</math>

In other words, the only functions that have "elementary antiderivatives" (that is, antiderivatives living in, at worst, an elementary differential extension of <math>F</math>) are those with this form.  Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions.

A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al.<ref>{{harvnb|Geddes|Czapor|Labahn|1992}}</ref> See Lützen's scientific bibliography for a sketch of Liouville's original proof <ref>{{Cite book |last=Lützen |first=Jesper |url=http://link.springer.com/10.1007/978-1-4612-0989-8 |title=Joseph Liouville 1809–1882 |date=1990 |publisher=Springer New York |isbn=978-1-4612-6973-1 |series=Studies in the History of Mathematics and Physical Sciences |volume=15 |location=New York, NY |doi=10.1007/978-1-4612-0989-8}}</ref> (Chapter IX. Integration in Finite Terms), its modern exposition and algebraic treatment (ibid. §61).

==Examples==

As an example, the field <math>F := \Complex(x)</math> of [[rational function]]s in a single variable has a derivation given by the standard [[derivative]] with respect to that variable.  The [[#Constants of a differential field|constants of this field]] are just the [[complex number]]s <math>\Complex;</math> that is, <math>\operatorname{Con}(\Complex(x)) = \Complex,</math>

The function <math>f := \tfrac{1}{x},</math> which exists in <math>\Complex(x),</math> does not have an antiderivative in <math>\Complex(x).</math>  Its antiderivatives <math>\ln x + C</math> do, however, exist in the [[#logarithmic extension|logarithmic extension]] <math>\Complex(x, \ln x).</math>

Likewise, the function <math>\tfrac{1}{x^2+1}</math> does not have an antiderivative in <math>\Complex(x).</math>  Its antiderivatives <math>\tan^{-1}(x) + C</math> do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions.  However, a calculation with [[Euler's formula]] <math>e^{i \theta} = \cos \theta + i \sin \theta</math> shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions).
<math display=block>\begin{align}
e^{2i \theta} & 
= \frac{e^{i \theta}}{e^{-i \theta}} 
= \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} 
= \frac{1 + i \tan \theta}{1 - i \tan \theta} \\
\theta & 
= \frac{1}{2i} \ln \left(\frac{1 + i \tan \theta}{1 - i \tan \theta}\right) \\
\tan^{-1} x & 
= \frac{1}{2i} \ln \left(\frac{1+ix}{1-ix}\right)
\end{align}</math>

==Relationship with differential Galois theory==

Liouville's theorem is sometimes presented as a theorem in [[differential Galois theory]], but this is not strictly true.  The theorem can be proved without any use of [[Galois theory]].  Furthermore, the [[Galois group]] of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration).  Thus, an antiderivative's [[differential Galois group]] does not encode enough information to determine if it can be expressed using elementary functions, the major condition of Liouville's theorem.
<!--
== Example of theorem ==
Suppose we want to know whether a function of the form f*e<sup>g</sup> has an elementary antiderivative, with <math>f</math> and <math>g</math> in <math>C(x)</math> -->

==See also==

{{Div col}}
* {{annotated link|Algebraic function}}
* {{annotated link|Closed-form expression}}
* {{annotated link|Differential algebra}}
* {{annotated link|Differential Galois theory}}
* {{annotated link|Elementary function}}
* {{annotated link|Elementary function arithmetic}}
* {{annotated link|Liouvillian function}}
* {{annotated link|Nonelementary integral}}
* {{annotated link|Risch algorithm}}
* {{annotated link|Tarski's high school algebra problem}}
* {{annotated link|Transcendental function}}
{{Div col end}}

==Notes==
{{reflist|2}}

==References==

* {{Citation | last1=Bertrand | first1=D. | title=Review of "Lectures on differential Galois theory" | url=https://www.ams.org/bull/1996-33-02/S0273-0979-96-00652-0/S0273-0979-96-00652-0.pdf | year=1996 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=33 | issue=2|doi=10.1090/s0273-0979-96-00652-0 | doi-access=free }}
* {{cite book | first1=Keith O. |last1=Geddes |first2=Stephen R. |last2=Czapor |first3=George |last3=Labahn | title=Algorithms for Computer Algebra | url=https://archive.org/details/algorithmsforcom0000gedd | url-access=registration | publisher=Kluwer Academic Publishers | year=1992 | isbn=0-7923-9259-0}}
* {{Cite journal
 | last = Liouville
 | first = Joseph
 | author-link = Joseph Liouville
 | title = Premier mémoire sur la détermination des intégrales dont la valeur est algébrique
 | journal = Journal de l'École Polytechnique
 | year = 1833a
 | volume = tome XIV
 | pages = 124–148
 | url = http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f127.item.r=Liouville
 }}
* {{Cite journal
 | last = Liouville
 | first = Joseph
 | author-link = Joseph Liouville
 | title = Second mémoire sur la détermination des intégrales dont la valeur est algébrique
 | journal = Journal de l'École Polytechnique
 | year = 1833b
 | volume = tome XIV
 | pages = 149–193
 | url = http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f152.item.r=Liouville
 }}
* {{Cite journal
 | last = Liouville
 | first = Joseph
 | author-link = Joseph Liouville
 | title = Note sur la détermination des intégrales dont la valeur est algébrique
 | journal = [[Journal für die reine und angewandte Mathematik]]
 | year = 1833c
 | volume = 10
 | pages = 347–359
 | url = http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002139332
 }}
* {{Citation | authorlink=Andy Magid | last1=Magid | first1=Andy R. | title=Lectures on differential Galois theory | url=https://books.google.com/books?id=cJ9vByhPqQ8C | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=University Lecture Series | isbn=978-0-8218-7004-4 | mr=1301076 | year=1994 | volume=7}}
* {{Citation | last1=Magid | first1=Andy R. | title=Differential Galois theory | url=https://www.ams.org/notices/199909/fea-magid.pdf | mr=1710665 | year=1999 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=46 | issue=9 | pages=1041–1049}}
* {{Citation | last1=van der Put | first1=Marius | last2=Singer | first2=Michael F. | title=Galois theory of linear differential equations | url=http://www4.ncsu.edu/~singer/ms_papers.html | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-44228-8 | mr=1960772 | year=2003 | volume=328}}

==External links==

* {{MathWorld|id=LiouvillesPrinciple|title=Liouville's Principle}}

[[Category:Differential algebra]]
[[Category:Differential equations]]
[[Category:Field (mathematics)]]
[[Category:Theorems in algebra]]